Задача №1095

\[ \lim_{x\to{1}}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1} =\left[\frac{0}{0}\right] =\left[\begin{aligned}& t=\sqrt[mn]{x};\\& t\to{1}\end{aligned}\right] =\lim_{t\to{1}}\frac{t^m-1}{t^n-1}=\\ =\lim_{t\to{1}}\frac{(t-1)\cdot\left(t^{m-1}+t^{m-2}+\ldots+1\right)}{(t-1)\cdot\left(t^{n-1}+t^{n-2}+\ldots+1\right)} =\lim_{t\to{1}}\frac{t^{m-1}+t^{m-2}+\ldots+1}{t^{n-1}+t^{n-2}+\ldots+1} =\frac{m}{n}. \]